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\section{\usemenu{slac-pub-7172::context::slac-pub-7172-0-0-1}{Introduction}}\label{section::slac-pub-7172-0-0-1}
Heavy quark ($Q$=$c$,$b$) systems provide important laboratories for
experimental tests of the
theory of strong interactions, quantum chromodynamics (QCD). Since
the large quark mass $M_Q$ acts as a cutoff for soft gluon radiation,
some properties of these systems can be calculated accurately in
perturbative QCD. In other cases, however, where QCD calculations
assume massless quarks, the
products of heavy hadron decays can complicate the comparison of data
with the predictions for massless partons. It is therefore
desirable to measure properties of both light- and heavy-quark systems
as accurately as possible.
In this paper we consider one of the most basic observable properties of
high energy particle interactions, the multiplicity of charged
hadrons produced in the final state. We consider
hadronic $Z^0$ decays, which are believed to proceed via creation of a primary
quark-antiquark pair,
$Z^0$ $\rightarrow$ $q\bar{q}$,
which subsequently undergoes a fragmentation
process to produce the observed jets of hadrons. If the primary
event flavor $q$ can be identified experimentally, one can measure
the average charged multiplicity $\aven_q$ in events of that flavor,
for example $q=b,c,uds$, where $uds$ denotes the average over events of the
types $Z^0$ $\rightarrow$ $u\bar{u}$, $d\bar{d}$, and $s\bar{s}$.
These are not only important properties of $Z^0$ decays, but,
if the average decay multiplicity of the {\it leading}
hadrons that contain the primary heavy quark or antiquark is subtracted from
$\aven_Q$ to yield the average {\it non-leading} multiplicity,
can also
be used to test our understanding of the quark fragmentation process and its
dependence on the quark mass.
The hypothesis of flavor-independent
fragmentation \cite{1,2} implies
that this non-leading multiplicity in
$e^+e^- \to Q\bar{Q}$ (``heavy quark'') events
at center-of-mass (c.m.) energy $W$
should be equal to the total multiplicity in e$^+$e$^- \rightarrow
u\bar{u}, d\bar{d}$, and $s\bar{s}$ (``light quark'') events at a
lower c.m. energy given by the average energy of the non-leading system,
$E_{nl}$ = $(1-\langle x_{E_Q}\rangle )W$, where
$\langle x_{E_Q}\rangle$ = $2\langle E_Q\rangle/W$
is the mean fraction of
the beam energy carried by a heavy hadron of flavor $Q$.
Perturbative QCD predictions have been made \cite{3}
of the multiplicity {\it difference} between heavy- and
light-quark events, $\Delta\aven_Q$ = $\aven_Q-\aven_{uds}$.
In this case the suppression of soft gluon radiation
caused by the heavy quark
mass leads to a depletion of the non-leading multiplicity, and
results in the striking prediction that $\Delta\aven_Q$ is independent of $W$
at the level of $\pm$0.1 tracks.
Numerical predictions of $\Delta\aven_b=5.5\pm1.3$
and $\Delta\aven_c=1.7\pm1.1$ were also given \cite{3}.
More recently, improved calculations have been performed \cite{4},
confirming that the
energy-dependence is expected to be very small and predicting
$\Delta\aven_b$=3.53$\pm$0.23 and $\Delta\aven_c$=1.02$\pm$0.24 at $W=M_{Z^0}$.
In our previous paper \cite{5} we measured $\aven_b$ and
$\Delta\aven_b$ using the sample
of about 10,000 hadronic $Z^0$ decays recorded by the SLD experiment
in the 1992 run. By comparing with similar
measurements at lower c.m. energies \cite{1,6,7,8}
we found that $\Delta\aven_b$
was consistent with an energy-independent value, and in agreement with
the prediction of \cite{3}.
This result was subsequently confirmed by the
DELPHI \cite{9} and OPAL \cite{10} Collaborations.
The dominant uncertainty in our
measurement resulted from lack of knowledge of the charged multiplicity
in $Z^0$ $\rightarrow$ $c\bar{c}$ events, $\aven_c$.
In this paper we present simultaneous
measurements of $\aven_b$, $\aven_c$ and $\aven_{uds}$
based upon the sample of
about 160,000 hadronic $Z^0$ decays collected by SLD between 1992 and
1995, and using the SLD micro-vertex detector and tracking system
for flavor separation.
By measuring $\aven_c$ and $\aven_{uds}$ directly we have reduced
the systematic uncertainty on $\Delta\aven_b$ substantially, and have also
derived $\Delta\aven_c$, which allows us to compare with the QCD
predictions for the charm system
and with the only other measurement of this
quantity \cite{10} at the $Z^0$ resonance.
This measurement supersedes our previous measurements of $\aven_b$ and
$\Delta\aven_b$ \cite{5}.
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% Apparatus and Hadronic Event Selection
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